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| Papoulis A. — The Fourier Integral and Its Applications |
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| Предметный указатель |
All-pass filter 97
All-pass filter, ideal 109
All-pass filter, rational 117
Amplitude distortion 95—108 (see also Filters)
Analytic functions 283—311
Analytic properties of causal transforms 214
Analyzer (see Spectrum analyzer)
Angelo, E. J., Jr. 124
Asymptotic forms, central-limit theorem 227
Asymptotic forms, Laplace 302
Asymptotic forms, lattice form of 230
Asymptotic forms, saddle-point 302—307
Asymptotic forms, stationary phase 139—143
Asymptotic slope (see Signal-front delay)
Autocorrelation, derivatives 251
Autocorrelation, finite energy signals 242
Autocorrelation, finite power signals 245
Autocorrelation, periodic 250
Autocorrelation, pulse train 249
Band-limited interpolation 52 161
Band-limited signals, optimal 67
Band-limited signals, sampling theorem 50
Bandwidth, dynamic 159
Bernoulli theorem 232
Bessel functions 115
Bessel functions, asymptotic forms 142 305
Bessel functions, transform of 184
Beta density 238
Binomial distribution 50
Binomial filter 101
Blackman, R. B. 145
Bochner, S. 9
Bochner’s theorem 225
Bode, H. W. 192
Bounded variation 9
Bounds on a signal 34
Br (Bromwich path) 175
Branch points 288 298
Branch-cut line 288
Bremmer, H. 187
Butterworth filter with linear phase 105
Butterworth filter, causal 108
Butterworth filter, energy of 219
Calculus of residues (see Residues)
Cauchy principal value 9
Cauchy — Riemann conditions 285
Cauchyв’s formula 295
Cauchyв’s integral theorem 292
Cauer, W. 307
Causal form of central-limit theorem 234
Causal functions 13
Causal functions, evaluation of 56
Causal systems 85
Causal transforms 195—217
Causal transforms, analyticity of 214
Causality conditions 213—217
Central-limit theorem 227—239
Central-limit theorem for time-limited signals 238
Central-limit theorem, causal form 234
Central-limit theorem, error correction 233 236
Central-limit theorem, lattice form 229
Chalk, J. H. H. 74
Characteristic functions 224
Characteristic functions, density 235
Churchill, R. V. 169
Complex integration 290
Conformality 285
Conjugate functions 16
Continuity 284
Continuity of autocorrelation 266
Convolution 26
Convolution and correlation 247
Convolution theorem, frequency 27
Convolution theorem, time 26
Convolution, evaluation of 59
Convolution, repeated 226
Copson, E. J. 140
Correlation 242—254 (see also Autocorrelation)
Corrington, M. S. 105
Cross-correlation, finite energy signals 244
Cross-correlation, finite power signals 252
Cross-correlation, periodic 254
Cross-energy spectrum 28 244
Cross-power spectrum 252
Davenport, W. B., Jr. 240
De Bruijn, N. G. 302
Delay time 96
Delay, group 113 135
Delay, phase 135
Delay, signal-front 135 187
Delta function 273—281
Delta function as a limit 279
Delta function, convolution of 275
Delta function, derivative of 274
Delta function, finite integral of 275
Delta function, transform of 36
Delta function, zero division of 39
density 223
Density, beta 238
Density, lattice 224
Detection 162
Differentiation, frequency 16
Differentiation, of autocorrelation 251
Differentiation, time 16
Discontinuities 9 30
dispersion 18 226 230
Distortion, amplitude 95—108
Distortion, phase 108—116
Distributions 270—281
Distributions, as generalized limits 278
Distributions, derivatives of 273
Distributions, physical concepts as 281
Distributions, properties of 272
Doetsch, G. 169
Duration of a signal, RMS 62
Duration of a signal, time-limited 66
Duration of a signal, with linear phase 64
Dynamic bandwidth 159
Echos 101 103
Eigenfunctions 68 83
Eigenfunctions for maximum energy 68—74
Eigenfunctions of linear systems 83
Eigenvalues 68 84
Energy of a signal 212
Energy spectrum 28
Energy theorem 27 194
Energy, finite, signals with 241—245
Erdelyi, A. 302
Error function 72 104
Even functions 11
Even functions, bilateral transforms of 190
Even-odd decomposition 12
Exponential, transform of 23
Extremum theorem 286
Fejer kernel 31
Fejer sum 46
Felsen, L. B. 302
Filters 94—136
Filters causal, with linear phase 106—108
Filters in-phase 132
Filters, all-pass (see All-pass filter)
Filters, bandpass 120—136
Filters, bandpass transformation 122 127
Filters, binomial 101
Filters, Butterworth (see Butterworth filter)
Filters, cosine 101
Filters, equivalent low-pass 122 132
Filters, gaussian 104
Filters, high-pass 100
Filters, ideal 97
| Filters, low-pass 94—119
Filters, low-pass, general 103
Filters, modulated input 127 133
Filters, narrow-band 126
Filters, pulse-forming 116 (see also Phase distortion)
Filters, quadrature 132
Filters, symmetrical 120——131
Filters, unsymmetrical 131—134
Fourier integral, causal 13
Fourier integral, even 11
Fourier integral, imaginary 11
Fourier integral, kernel 18 30
Fourier integral, numerical techniques 53—59
Fourier integral, odd 11
Fourier integral, proof of 29
Fourier integral, real 11
Fourier integral, special forms 10—13
Fourier series 42—47
Fourier series, kernel 44
Fourier transform, generalized 259 261
Fourier transform, running 148 (see also Fourier integral)
Frequency modulation 165
Frequency shifting 162
Fresnel integrals 164 165 218
Friedman, B. 269
Functional 270 (see also Distributions)
Gain-bandwidth theorem 218
Gamma function 171 220
Gaussian filters 104
Gaussian function in central limit 224
Gaussian function, transform of 24
Generalized derivatives 276
Generalized Fourier transform 259 261
Generalized functions 270
Generalized harmonic analysis 259—264
Generalized harmonic analysis and power spectrum 261
Gibbs’ phenomenon 30
Gnedenko, B. V. 227
Goldman, S. 127
Group delay 134
Group delay, Fourier expansion of 113
Group velocity 139
Guillemin, E. A. 14
Halperin, I. 259
Harmonic analysis 259—264
Hermite polynomials in central limit 233
Hermite polynomials, transform of 78
High-pass filter 100
Hilbert transforms 198—212
Hurwitz polynomials 195 205
Hurwitz polynomials from biquadratic factors 206
Ideal filter, all-pass 109
Ideal filter, bandpass 123
Ideal filter, low-pass 97
Impulse function 269—282 (see also Delta function)
Impulse response of linear systems to 82
In-phase filter 132
Initial-value theorem 186
Input 81
Input wide-band 128 133
Input, modulated 127—134
Input, narrow-band 127 133
Instantaneous power spectrum 149
Integral equation 68
Integral theorems 192—218
Integration, complex 290
Interpolation, bard-limited 52 161
Inversion formula, evaluation of 56
Inversion formula, Fourier 7
Inversion formula, proof of 29
Jordan’s lemma 300
Kernel, Fejer 22 31
Kernel, Fourier integral 18 30 280
Kernel, Fourier series 44
Kharkevich, A. A. 145
Knopp, K. 283
Kolmogorov, A. 227
Kupfmiiller, K. 94
Laguerre polynomials 203 236
Laguerre polynomials, error correction with 236
Laguerre polynomials, expansion in 203
Landow, H. T. 67
Laplace transform 169—191
Laplace transform, analyticity of 170
Laplace transform, bilateral 187—191
Laplace transform, bilateral, of even functions 190
Laplace transform, branch points 182—185
Laplace transform, evaluation of inverse 176—185
Laplace transform, finitely many poles 176—179
Laplace transform, infinitely many poles 179
Laplace transform, inversion of 175
Laplace transform, periodic functions 181
Laplace transform, region of existence 170
Laplace transform, relationship with Fourier transform 172—175
Laplace transform, unilateral 169—187
Laplace, asymptotic method of 302
Laplace, bilateral 188
Laplace, unilateral 175 (see also Laplace transform)
Lattice densities 224
Lee, Y. W. 240 (see also Wiener-Lee transforms)
Lighthill, M. J. 269
Limit theorem (see Central-limit theorem)
Limits, generalized 277—281
Line integral 290
Line power spectra 250
Line spectra 42—47
Line spectra, autocorrelation with 249
Linear phase causal systems 106—108
Linear phase filters 95—106
Linear phase signals, duration of 64
Linear systems 82
Linear systems, causal 85
Linear systems, output of 82
Linear systems, time-invariant 83
Linearity 14 82
Lommel’s Bessel formula 115
low-pass filters (see Filters)
Lukacs, E. 224
Markuvitz, N. 302
Mean 18
Mean for pulse train 230
Mean, evaluation of 226
Minimization of signal duration 63
Minimization, RMS 33
Minimum-phase-shift functions 204—212
Minimum-phase-shift functions, Hilbert transforms 206—209
Modulated signals (see Input modulated)
Modulation in spectral analysis 150
Modulation, frequency 165
Moment theorem 16—18
Moments 16
Moments, evaluation of 226
Monotonic response conditions 90
Multivalued functions 287 299
Multivalued functions, inverse transform of 182—185
Murakami, T. 105
Noise, thermal 266
Numerical techniques 53—61
Operator, linear 82
Optimum, all-pass filter 163
Optimum, input 73
Output 81
Output of linear systems 82
Overshoot 90
Page, C. H. 149
Paley — Wiener condition 215—217
Parseval’s formula 27
Parseval’s formula for causal functions 194
Partial sum 46
Periodic input to linear systems 86
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